TITLE: Simply connected does not imply contractible. Is there a nice counter example in $R^2$?
QUESTION [25 upvotes]: The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were interested in a counter example in the plane - does anyone know of a subset of $R^2$ which is simply connected but not contractible?

REPLY [6 votes]: John's answer is absolutely correct, but here is an addendum to it: All proper subsets $Z$ of the plane are aspherical in the sense that every continuous map 
$$
f: S^n\to Z, n>1,
$$
extends to a continuous map of the ball, $B^{n+1}\to Z$. See the paper One-dimensional sets and planar sets are aspherical by Cannon, Conner and Zastrow.